Properties

Label 4-4312-1.1-c1e2-0-2
Degree $4$
Conductor $4312$
Sign $1$
Analytic cond. $0.274936$
Root an. cond. $0.724116$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 4·7-s − 3·8-s − 2·9-s − 11-s + 4·14-s − 16-s − 2·18-s − 22-s − 8·23-s − 6·25-s − 4·28-s + 4·29-s + 5·32-s + 2·36-s + 44-s − 8·46-s + 9·49-s − 6·50-s − 8·53-s − 12·56-s + 4·58-s − 8·63-s + 7·64-s + 20·67-s + 12·71-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s − 2/3·9-s − 0.301·11-s + 1.06·14-s − 1/4·16-s − 0.471·18-s − 0.213·22-s − 1.66·23-s − 6/5·25-s − 0.755·28-s + 0.742·29-s + 0.883·32-s + 1/3·36-s + 0.150·44-s − 1.17·46-s + 9/7·49-s − 0.848·50-s − 1.09·53-s − 1.60·56-s + 0.525·58-s − 1.00·63-s + 7/8·64-s + 2.44·67-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4312\)    =    \(2^{3} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.274936\)
Root analytic conductor: \(0.724116\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9950697187\)
\(L(\frac12)\) \(\approx\) \(0.9950697187\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.27021067188370396709985028906, −11.99297690089808119582251325551, −11.33696894235438898578899248708, −10.89556905826929887808213229599, −9.955763757492334327628645561266, −9.520825901634028793940380124781, −8.509739114195429978253782788936, −8.214170704131043524266746631207, −7.70358982130509525195216328506, −6.48020125855069284363283598797, −5.74125687391337059330576407197, −5.15684451224741078245187981572, −4.43684984520239592288748109335, −3.62979773637590352656751406779, −2.24772405847854638205107848265, 2.24772405847854638205107848265, 3.62979773637590352656751406779, 4.43684984520239592288748109335, 5.15684451224741078245187981572, 5.74125687391337059330576407197, 6.48020125855069284363283598797, 7.70358982130509525195216328506, 8.214170704131043524266746631207, 8.509739114195429978253782788936, 9.520825901634028793940380124781, 9.955763757492334327628645561266, 10.89556905826929887808213229599, 11.33696894235438898578899248708, 11.99297690089808119582251325551, 12.27021067188370396709985028906

Graph of the $Z$-function along the critical line